Models for electrostatic drift waves with density variations along magnetic field lines

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Models for electrostatic drift waves with density variations along magnetic ﬁeld lines

O. E. Garcia¹and H. L. Pécseli²

Received 26 August 2013; revised 10 October 2013; accepted 11 October 2013; published 5 November 2013.

[1] Drift waves with vertical magnetic fields in gravitational ionospheres are considered where the unperturbed plasma density is enhanced in a magnetic flux tube. The gravitational field gives rise to an overall decrease of plasma density for increasing altitude. Simple models predict that drift waves with finite vertical wave vector components can increase in amplitude merely due to a conservation of energy density flux of the waves. Field-aligned currents are some of the mechanisms that can give rise to fluctuations that are truly unstable. We suggest a self-consistent generator or

“battery” mechanism that in the polar ionospheres can give rise to magnetic field-aligned currents even in the absence of electron precipitation. The free energy here is supplied by steady state electric fields imposed in the direction perpendicular to the magnetic field in the collisional lower parts of the ionosphere or by neutral winds that have similar effects.Citation: Garcia, O. E., and H. L. Pécseli (2013), Models for electrostatic drift waves with density variations along magnetic field lines, Geophys. Res. Lett., 40, 5565–5569, doi:10.1002/2013GL057802.

1. Introduction

[2] In nature as well as many laboratory experiments, we have conditions where the equilibrium plasma density has a gradientalong as well asperpendicularto an externally imposed stationary magnetic fieldB. In the polar ionosphere, in particular, magnetic flux tubes with enhanced plasma density are often associated with auroral precipitation and auroral patches [Hosokawa et al., 2010] where the plasma density has a gradient? Bas well as a decreasing density along magnetic field lines, i.e., increasing altitudes. The case with plasma density gradients perpendicular toB has pre- viously been studied, and conditions for electrostatic drift wave instabilities have been established [Kadomtsev, 1965].

The basic results are generalized here to allow for a density gradient along the magnetic field as well, using a simple solvable gravitational model. The real and the imaginary parts of the dispersion relation of the waves are modified by this vertical gradient. For kinetic collisionless plasma conditions, we find that the most likely source of enhanced growth rates is a field-aligned electron current. We propose

1Department of Physics and Technology, University of Tromsø, Tromsø, Norway.

2Department of Physics, University of Oslo, Oslo, Norway.

Corresponding author: H. L. Pécseli, Department of Physics, University of Oslo, Box 1048 Blindern, NO-0316 Oslo, Norway.

([email protected])

0094-8276/13/10.1002/2013GL057802

a novel generator (or “battery”) mechanism relevant for polar ionospheres.

2. Drift Wave Instability

[3] We consider a horizontally stratified plasma in a gravitational fieldg = –gb^zand vertical magnetic fieldB = Bb^z. The basic equations for the plasma assume cold ions and retain the polarization drift as a correction to theEB/B² ion velocity. A steady state solution is found asn(r?,z) = n0(r?) exp(–z/LV)for the density with a vertical length scale LV Te/Mg C²_s/g. For the electrostatic potential, we have (z) = –zMg/e. With a constant electron tempera- tureTe, the Brunt-Väisälä frequency [Yeh and Liu, 1972] is _BV = g/Cs, whereCs = p

T_e/M is the sound speed for cold ions, introducingMas the ion mass. For relevant conditions, we can haveCs 500–1000m s^–1, giving_BV 0.01–0.02 s^–1. We have n0(r?) accounting for the density enhancement in the magnetic flux tube, while the exponential term exp(–z/LV)gives the overall density decrease for increasing altitudezrepresenting a balance between the gravitational force on the ions and the constant vertical ambipolar electric field componentE = (Mg/e)b^zresulting from the electron pressure. The gravitational field is here used as giving a solvable, yet representative, model for general mechanisms that give rise to plasma density gradients along magnetic field lines. For the ionosphericFregion, we can use an approximation with an exponentially decreasing plasma density for altitudes above theFmaximum.

[4] Starting from the linearized ion continuity and ion momentum equations with cold ions, Ti 0, we obtain after some straightforward algebra an equation for the ion dynamics that gives a dynamic linear relation between the ion density ni with ni/n and the electrostatic potentialas

@²

@t²– 1 Bci

@²r_?²

@t² –

@r?/@tB

B² + 1

Bci

@²r?

@t²

r?lnn₀

– e M

@

@z

@

@zlnn– e M

@²

@z² = 0 . (1)

The set of equations are closed by the assumption of quasi- neutrality and an equation relatingandobtained from the electron dynamic equations [Chen, 1984]. The simplest such relation assumes electrons to be isothermally Boltzmann distributed at all times. For a plane wave solutionexp(–i(!t– kr)), we ﬁnd the dispersion relation!=!(k)in the form

!²

1 +k²_?a²_i

–C_s!(a_ik?)b^z^r^?^lnⁿ0

–igk_z–C²_sk²_z= 0 , (2)

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witha²_i Te/(eBci) =C²_s/²_ci. We tookk? ? r?n. The correction due to the ion polarization drift is found in the (1 + k²_?a²_i) multiplier on !². The z variation cancels in the drift frequency deﬁned as!^*k?(Te/eB)|rlnn0(r?)|, making it independent of altitude. With this standard expres- sion [Kadomtsev, 1965] for the drift frequency!^*, we ﬁnd

!(k?,k_z) = !^*˙ q

!^*2+ 4

1 +a_i²k_?² C_s²k_z²+igk_z 2

1 +a_i²k?2 , where we assumek to be real. We recognize the acceler- ated and decelerated ion sound waves [Kadomtsev, 1965], generalized here by an imaginary part originating from the B-parallel density gradient. For upward propagating waves, the relation (2) has solutions which grow exponentially with time. Waves propagating downward (!/kz< 0) are damped.

For kz = 0, we have ! = !^*, while for k? = 0, we have !^* = 0and the acoustic plasma gravity waves with

! = ˙p Cs2

kz2

+igkz. The two wave modes are coupled when bothk?¤0andkz¤0.

[5] An initial value problem (with real k) as outlined before is relevant for instance for numerical simulations, but for physical applications, for instance in a laboratory plasma, a boundary condition is more appropriate, where now the applied frequency is real and a complex wave vector accounts for spatial damping or growth of the waves. With the boundary conditionexp(–i!t+ik?x)atz = 0also this problem can be solved forkz(!,k?)giving

k_z(!,k_?) = – ig˙

q 4C_s²

!²

1 +a_i²k?2 –!^*!

–g²

2C_s² , (3)

which can be written as k_zL_V= –i

2˙1 2

q

4(!²(1 +k²_?a²_i) –! !^*)/²_BV– 1.

The normalized result (3) is illustrated in Figure 1. The result contains a ratio of length scales that is here assigned a numerical value`/L_V = 0.2 where`^–1 d lnn0/dz. For

!²(1 +k²_?a²_i) –! !^*<²_BV/4, we ﬁnd a stopband, where the wave has no spatial oscillations along the verticalz direction, only an exponential form that does not correspond to propagating waves with real phase velocity.

[6] The observed wave growth cannot properly be called an instability [Parkinson and Schindler, 1969;Dysthe et al., 1975;D’Angelo et al., 1975]. The amplitude increase with increasing altitudezis merely a consequence of the decreasing plasma density: For fluctuating plasma velocitiesu, the vertical component of the wave energy density flux is to lowest ordern(r?,z)Mu²Cs. The sound velocityCsis independent of plasma density, so to have a constant vertical energy flux, we must haveu²increasing whennis decreasing and vice versa. For!²(1 +k²_?a²_i) –! !^* > ²_BV/4, we thus have={kzLV} = ¹₂, in which case the energy flux den- sityCsnMu² is independent ofzas stated. Since this is not a proper instability driven by free energy, we use the term

“geometric growth.” An observer will, however, ﬁnd that the ﬂuctuation level increases with altitude and might interpret this as an instability. When the wavelength becomes smaller than the vertical length scaleLV, the relative growth rate ={!}/<{!} becomes small, and the waves can be assumed to propagate with the local dispersion relation, corresponding to the appropriate altitudez.

Figure 1. Normalized dispersion relation for real applied frequency!/BV and complex wave number kzLV, where

<{}and={}denote real and imaginary parts, respectively, of the term in the angular brackets. Negative ={kz} corresponds to spatial growth in the positivezdirection.

[7] Allowing for a nonvanishing ion temperature, TiTe, we ﬁnd in the quasi-neutral limit that ion Landau damping with g = 0 gives an imaginary part of the frequency so that the ratio of the real and imaginary parts are constant for varying real wave number, when an initial value problem is considered. For a similar boundary value problem, the ratio of imaginary and real wave numbers is approximately constant for varying real frequency. For the geometric growth discussed for the initial value problem with real wave numbers, we found that the imaginary part of the frequency approached a constant level for largekz, i.e.,={!}/<{!}!0forkz! 1. It was demonstrated by Parkinson and Schindler[1969] that for stable conditions, the ion Landau damping will dominate for largekz.

[8] To generate a substantial perturbation consistent with observations [Greenwald et al., 2002; Hosokawa et al., 2010], free energy is needed to drive an instability. Several sources of energy can be recognized as relevant for the present problem. Here we focus on electron kinetic effects.

These contributions to the instabilities are important for conditions where the growth rates are comparable to or exceed- ing characteristic ion collision frequencies. For more general cases, modiﬁcation due to collisions needs to be included.

[9] In order to include electron kinetic effects, we relax the condition on Boltzmann-distributed electrons and use a drift kinetic electron model [Kadomtsev, 1965].

@f

@t+r?(uEBf) +u_k@f

@z+ e m

@

@z

@f

@uk

= 0 , (4)

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withm being the electron mass, uEB –r? B/B², andf = f(z,r?,uk,t) is here the electron gyrocenter distribution function. The polarization drift is ignored for the electrons. Since the magnetic momentmis constant, within the relevant approximations,mdoes not appear explicitly in (4).

[10] We ﬁnd a steady state velocity distributionffrom –r?B

B² r?f+uk

@

@z f+ e m

@

@z

@

@uk

f= 0 . (5) We anticipated a constant vertical electric ﬁeldE= (Mg/e)b^z to compensate the (constant) gravitational force on the ions.

For steady state, we haver?= 0. We readily obtain f(x,z,u_k)n₀(x) exp(–zMg/Te)F(u_k)n(z)F(u_k) , (6) where F(uk) is a Maxwellian with normalization R¹

–1F(uk)duk= 1. Deviations from this form will in general imply modiﬁcations of the steady state. Such changes are generally small. We here used a local model for the modi- ﬁcations of the growth rate of the drift wave instability as caused by deviations from strictly isothermally Boltzmann- distributed electrons and considered again the initial value problem with real k. The local model is applicable when

!>_BVgivingkzLV> 1, see Figure 1 for<{k_zLV}. [11] For a slowly drifting Maxwellian electron velocity distribution with a drift velocity U uTe, where uTe = pTe/mis the electron thermal velocity, we obtain by (4) a relation between plasma density and potential in the form

ne

n = e Te

!–Ukz–!^* k_zu_T_e Z

!–Ukz

k_zu_T_e

+ 1

, (7)

whereZ is the plasma dispersion function. In this kinetic model, (7) is replacing the assumption of Boltzmann- distributed electrons. In, for instance, laboratory experiments, it has been demonstrated that drift waves can be made unstable by currents along the magnetic ﬁeld lines [Hatakeyama et al., 2011]. To estimate the growth rate under the present conditions, we again assume quasi-neutrality neni=nand combine (1) and (7) to get

!^*

! –C²_s k²_? ²_ci – k²_z

!²

! +ik_zg

!² = 1 +!–Uk_z–!^* k_zu_T_e Z

!–Uk_z k_zu_T_e

. (8) [12] The result is local in the sense that the electron drift velocity U is taken to be constant. We have two limiting cases:! 0for the slow sound mode and! !^*for the fast mode. For small!/kz andU uTe, we can approxi- mateZ ip

. For small kz, we ﬁnd the previous results with realkand complex!for the slow mode (where!0) with negligible kinetic effects, while the fast mode (where

!!^*/(1 + (k?ai)²)) becomes

! !^*

1 + (k?a_i)²+C²_sk²_z

!^*

1 + (k?ai)²

+i kzg

!^*

1+(k?a_i)² +!^*p

Uk_z

1+ (k?a_i)²

+!^*(k?a_i)² k_zu_T_e

1 + (k?a_i)²2

! . (9) [13] Kinetic effects give rise to instabilities even when U = 0, but the growth rate ={!} is then small and scales with(k?ai)². Signiﬁcant growth rates can be obtained even

for moderate bulk electron ﬂow velocities, where waves are unstable in the direction determined by the condition Ukz> 0. The growth rate increases withUwith no threshold.

The contribution from the kinetic instability, i.e., the second term in the parenthesis in (9), will dominate the ﬁrst geometric growth term, whenU> uTekzg/(!^*)², in the limit of small(k?ai)².

[14] We have similar, but more lengthy, results for the boundary value problem with real applied frequency ! and complexkz as in Figure 1. The stopband is recovered also with these kinetic effects, but the vertical growth rate receives a contribution from a “true” instability.

3. Proposal for a Battery Mechanism

[15] A complete model has to account for electron current generators. Currents can for instance be caused by diffuse auroral electron precipitation [Ossakow and Chaturvedi, 1979], but these will generally not be localized at the density gradients. Here we focus on a different and more relevant current-generating mechanism, originating from steady state horizontal electric fieldsE0 imposed perpendicular toBin the ionospheric E region, demonstrating that a magnetic flux tube with enhanced density extending throughEandF regions will be unstable withE0 ¤ 0. We introduced the notationE0to distinguish this horizontal electric field from the vertical E. The electron and ion mobilities are different in the E region since !ce en while ci in in terms of electron and ion collision frequencies (en andin) with the neutral background gas. These neutral collision frequencies decrease rapidly with altitude. For altitudes above some 120–130 km, the electrons and ions are both drifting with essentially the sameE0 B/B² velocity, see for instance the illustration by Dyrud et al.[2006, Figure 3].

We model theEregion as a collisional horizontal “slab” of thicknessd with!ce en while ci in. For higher altitudes above approximately 120 km, we ignore collisions all together: As what concerns the steady state electron-ion velocity difference, this is a good approximation [Dyrud et al., 2006].

[16] The electrons ﬂow approximately at theE0 B/B² velocity, but the neutral drag on the ions implies that electron and ion steady state drifts differ to give the electrojet current [Primdahl and Spangslev, 1977;Kelley, 1989]. The model for the current generation is also understood in theE0B moving frame, where the neutral component is in motion.

The localized density enhancement can also be moving. To interpret the free energy driving the current as an electric ﬁeldE0 or a neutral wind is merely a question of choos- ing the frame of reference. With the present approximations, the electrojet current density isJ0 –en0E0/Bgiving a net currentI0–den0E0/Bper length unit alongE₀.

[17] If we have a local plasma density enhancement with a steady state density gradient perpendicular to B with r?n0 k E0Bin the rest frame as in Figure 2, we have a local enhancement of the net current in that region since E0is imposed externally to give a constant velocity and the number of charge carriers is locally enhanced. The electron currentI0+I1in the enhanced density region (n0+ nbetween the two intervalsa andb in Figure 2) is not compensated at the boundaries of the magnetic ﬂux tube with enhanced plasma density: In regionsaandb, the currentI₁therefore has to expand along the vertical magnetic ﬁeld lines as illus-

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Figure 2. Illustration of a cut through a magnetic ﬂux tube with enhanced plasma density. (top) Details of the geometry and density variations; (bottom) illustration of the currents generated by a steady state electric ﬁeld E0 ? B. The thickness of theEregion isd10–20km.

trated in Figure 2. It cannot escape downward into the D region because of the high collisionality there with a corresponding low electron mobility. In the vertical direction along magnetic field lines into theFregion, we can have a current propagating [Primdahl and Spangslev, 1977;Prim- dahl et al., 1987]. This current will flow along magnetic field lines perpendicular tor?n0 and give rise to unstable drift waves.

[18] The currentI1in Figure 2 is carried by the electrons, both in theEandFregions. With a horizontal density variation, we have an asymmetry between the part facing the E0B drift and the opposite side. Due to the abundance of electrons in theEregion, a quasi steady state is achieved rapidly where electrons are ﬂowing from the high plasma densityEregion into the lower density in theFregion. On the side facing theE0Bdrift in Figure 2 (regiona), the electrons have to ﬂow from a lower plasma density into the larger density in theEregion. In this latter case, stationary conditions will need longer time to be established. In theb region in Figure 2, the electron drift will enhance upward traveling low-frequency waves as described by (9); in region a, unstable waves are propagating in the opposite direction.

Standard models for the current convective instability in the Fregion [Ossakow and Chaturvedi, 1979] do not distinguish the direction of the electron ﬂow.

[19] The conservation of net current(I0+I1)through the cut between regions in Figure 2 (a andb) will act as an amplification for current densities and thereby average electron flow velocities. For regions outside the flux tube with enhanced plasma density, we have a net electrojet current I0 = J0d = en0dE0/B per unit length in the direction perpendicular tor?n0, so by Kirchhoff’s laws, we have for instance at the region b in Figure 2 where J1 is the current density, thated nE0/B J1b = be(n0+ n)U, giving the vertical electron drift velocity estimateU ( n/(n0+ n))(d/b)E0/B, so that U increases linearly with E0 with no threshold. In the ionosphericE region, we often have E0 20mV/m [Kelley, 1989] which givesE0/B Cs, so we can argue that substantial electron drifts can be achieved by this mechanism. Taking a reference case of`/LV = 0.1 andUCs, we find that the kinetic effects contribute with a growth rate of={!}/BV 0.1to the geometric growth over a wide wave number range. The results are sensitive to the parameter values: smaller`/LVgenerally leads to larger dominant frequencies and higher growth rates.

4. Conclusion

[20] The analysis outlined in this study has several ver- ifiable features that distinguishes it from, for instance, the current convective instability [Ossakow and Chaturvedi, 1979] that assumes a fully collisional plasma. These instabilities are often invoked for explaining observations of irregularities in the ionosphericEandFregions [Greenwald et al., 2002; Hosokawa et al., 2010]. While most studies of convective drift instabilities [Simon, 1963;Fejer et al., 1984] rely on steady state electric fields having a component alongr?n0so thatE0rn0> 0, our generator model is based on steady state electric fieldsE0 ? r?n0(r?)and operates self consistently with the density enhancement in a magnetic flux tube that forms the basis for a universal drift wave instability. The model predicts a difference in the characteristics of the fluctuations on the density gradient facing the electrojet current as compared to the one on the opposite side.

To the lowest order, the growth rate is directly proportional to the electron flow velocity as indicated by (9), where our previous arguments giveUproportional toE0/B. The analysis used a simplified gravitational model that can be solved directly but the basic physical arguments for the imaginary part of (2) are robust and apply also for cases where density gradients along magnetic fields are caused by sinks and sources [D’Angelo et al., 1975]. A complete description of the battery mechanism discussed in this communication requires modeling the current closure, which for ionospheric conditions implies a study of the full inhomogeneous magnetic field configuration [Primdahl and Spangslev, 1977, 1983]. This analysis is outside the scope of the present study.

[21] Acknowledgments. Parts of this work were carried out while one of the authors (HLP) was visiting the University of Tromsø. The ﬁnancial support and kind hospitality of this institution are gratefully acknowledged.

[22] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.

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